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62 HI METRIC SPACES

(3.17.1) Any precompact set is bounded.

This follows from the fact that a finite union of bounded sets is bounded
(3.4.4).

The converse of (3.17.1) does not hold in general, for any distance is
equivalent to a bounded distance (Section 3.14, Problem 2) (but see (3,17.6)).

(3.17.2) Any compact set in a metric space is closed.

Indeed, such a subspace is complete by (3.16,1), and we need only apply
(3.14.4).

(3.17.3) In a compact space E, every closed subset is compact.

For such a set is obviously precompact, and it is a complete subspace by
(3.14.5).

A relatively compact set in a metric space E is a subset A such that the
closure A is compact.

(3.17.4) Any subset of a relatively compact (resp. precompact) set is relatively
compact (resp. precompact).

This follows at once from the definitions and (3.173).

(3.17.5) A relatively compact set is precompact. In a complete space, a
precompact set is relatively compact.

The first assertion is immediate by (3.17.4). Suppose next E is complete
and A c E precompact. For any e > 0, there is a covering of A by a finite
number of sets C_fc of A having a diameter <a/2; each C_k is contained in
a closed ball D_fc (in E) of radius e/2. We have therefore A c (J D_A, the set

(J D_fc being closed, and each D_fc has a diameter < e. On the other hand, A is a
complete subspace by (3.14.5), whence the result.

A precompact space E which is not complete gives an example of a
precompact set which is not relatively compact in E.



	62 HI METRIC SPACES (3.17.1) Any precompact set is bounded. This follows from the fact that a finite union of bounded sets is bounded (3.4.4). The converse of (3.17.1) does not hold in general, for any distance is equivalent to a bounded distance (Section 3.14, Problem 2) (but see (3,17.6)). (3.17.2) Any compact set in a metric space is closed. Indeed, such a subspace is complete by (3.16,1), and we need only apply (3.14.4). (3.17.3) In a compact space E, every closed subset is compact. For such a set is obviously precompact, and it is a complete subspace by (3.14.5). A relatively compact set in a metric space E is a subset A such that the closure A is compact.

	(3.17.4) Any subset of a relatively compact (resp. precompact) set is relatively compact (resp. precompact). This follows at once from the definitions and (3.173).

	(3.17.5) A relatively compact set is precompact. In a complete space, a precompact set is relatively compact. The first assertion is immediate by (3.17.4). Suppose next E is complete and A c E precompact. For any e > 0, there is a covering of A by a finite number of sets C_fc of A having a diameter <a/2; each C_k is contained in a closed ball D_fc (in E) of radius e/2. We have therefore A c (J D_A, the set (J D_fc being closed, and each D_fc has a diameter < e. On the other hand, A is a complete subspace by (3.14.5), whence the result. A precompact space E which is not complete gives an example of a precompact set which is not relatively compact in E.